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Results tagged with big-list
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user 6794
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
1
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Equivalence relations not associated with a group
Since this question has been bumped to the front page, let me add Turing equivalence of subsets of $\mathbb N$. (Two such subsets are Turing equivalent iff each is computable by a Turing machine equip …
20
votes
Noteworthy, but not so famous conjectures resolved recent years
Maryanthe Malliaris and Saharon Shelah proved that the cardinal characteristics $\mathfrak p$ and $\mathfrak t$ are equal, answering a question that goes back at least to the 1970's and probably (with …
4
votes
Math talk for all ages
I like to show how the same mathematics shows up in very different contexts. A topic that I've used with quite varied audiences (though never with as much variation in a single audience as you have) i …
6
votes
Mathematicians with both “very abstract” and “very applied” achievements
Dana Scott's achievements include work in pure set theory and also work in computer science. He proved that there are no measurable cardinals in Gödel's constructible universe and (with Solovay) devel …
21
votes
Nonequivalent definitions in Mathematics
An extension of a group $A$ by a group $B$ can be either a group $G$ with a normal subgroup isomorphic to $B$ with $G/B$ isomorphic to $A$ or a group $G$ with a normal subgroup isomorphic to $A$ with …
10
votes
Important results with one or more than one proof
The first example that occurs to me is Hindman's theorem: If the set of positive integers is partitioned into finitely many pieces, then there is an infinite set $H$ such that all sums of finitely man …
31
votes
Mathematicians who made important contributions outside their own field?
Paul Cohen was an analyst but got a Fields Medal for his work in set theory, proving the independence of the continuum hypothesis from ZFC and the independence of the axiom of choice from ZF.
21
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Structures that turn out to exhibit a symmetry even though their definition doesn't
Consider the Desargues configuration. It consists of (1) two triangles, say $ABC$ and $A'B'C'$ such that the lines $AA'$, $BB'$, and $CC'$ all meet at a point $P$, and (2) the three points of intersec …
6
votes
Classic applications of Baire category theorem
I got some mileage out of the Baire category theorem in abelian group theory, in particular in the study of the group $\mathbb Z^{\aleph_0}$, the additive group of sequences of integers (which, contra …
3
votes
How should the Math Subject Classification (MSC) be revised or improved?
My experience has been that the editors of Math Reviews pay attention to suggestions about revisions of the classification system. I'm not saying that they implement all the suggestions (especially b …
22
votes
Examples of theorems with proofs that have dramatically improved over time
I described an example, Hindman's theorem, at https://mathoverflow.net/questions/94546 . The short version is that Hindman's original proof was unpleasantly complicated, whereas a later proof by Galv …
5
votes
Fundamental Examples
Motivated by Amit Kumar Gupta's answer about the continuum hypothesis, let me add an example that is less natural but has inspired an amazing amount of set theory, namely Suslin's Hypothesis. This co …
0
votes
Mathematical ideas named after places
The Arctic Circle Theorem (http://arxiv.org/abs/math/9801068)
1
vote
Mathematical ideas named after places
The Conway-Paterson-Moscow theorem
9
votes
Mathematical ideas named after places
anarboricity of graphs (named in honor of the city of Ann Arbor by Frank Harary, but also having something to do with non-trees (http://mathworld.wolfram.com/Anarboricity.html)